1. Field of the Invention
The present invention relates to a method for controlling the pH and other concentration variables in a chemical reactor, an analogous apparatus, a tank or similar chemical system, particularly when at least one desired outlet concentration or a function dependent on outlet concentrations is given.
2. Description of the Prior Art
In industry, nature, the technological environment, and elsewhere there appear processes in connection with which the concentrations of the various components in the material to be treated change under the effect of chemical reactions or comparable physical phenomena. Some such analogous phenomena are dissolving, crystallization, and adsorption, when they are comparable to chemical reactions in regard to their mathematical treatment.
Rate of reaction. The rate at which the participating components are formed or spent is determined by the kinetic dependence characteristic of the reaction or respective phenomenon, by the concentrations of the components participating in the reaction, and by other variables and parameters, such as temperature, pressure, and the properties and concentration of the catalyst. The rate r of a general irreversible reaction (1) is expressed, subject to certain limitations, by the dependence (2). The reaction formula (1) also links the quantitative changes in the various components to each other. EQU aA + bB + . . . .fwdarw. products of reaction (1) EQU r = kC.sub.A.sup.a C.sub.B.sup.b . . . (2)
a,b . . . number of mass units (in moles) of component A, B, . . according to the basic reaction formula, PA1 k rate coefficient of the reaction (depends on other process variables and parameters through which the progress of the reaction can be affected), PA1 r rate of reaction (the rate of conversion of the component under observation in moles per volume unit), PA1 C.sub.A,C.sub.B . . . concentration of component A, B . . . (in moles per volume unit). PA1 C.sub.in inlet concentration PA1 t,.theta. time variables
The mathematic expression (2) expresses the rate of reaction at the point where the values of the concentrations and other process variables and parameters are given. It can also be used to describe the operation of a batch reactor provided with perfect mixing. In that case, representation in the form of a differential equation is obtained for the concentration of each component. This equation can generally be solved in cases corresponding to the various orders of kinetics [1]. FNT [1] Campbell, D. C.: Process dynamics, Wiley 1958, pp 256-7.
When considering a kinetic continuous flow reactor, a reaction rate term is added to the differential equation or partial differential equation expressing the momentary material balance at each point. A general method for solving the concentration of the component under consideration at different points in the reactor, including the outflow point, has not been presented. Some special cases are, however, mastered. If, for example, the reaction occurring in a continuous flow reactor is of first-order kinetics, the process is linear in regard to the concentration, and the kinetic term in itself does not complicate the solving of the equation. Owing to the linearity the superposition principle is valid and the kinetic term can be added to the flow and mixing characteristics, the latter having first been solved or determined experimentally [2]. FNT [2] Levenspiel, O.: Chemical reaction engineering, Wiley 1962, pp 254-9, 266, 282, 289-93.
A reversible reaction (3) proceeds simultaneously from the left to the right and from the right to the left, and each partial reaction has its own rate. The gross rate is obtained as the difference of the individual rates (4). EQU aA + bB + . . . = mM + nN + . . . (3) EQU r = k.sub.1 C.sub.A.sup.a C.sub.B.sup.b . . . -k.sub.2 C.sub.M.sup.m C.sub.N.sup.n . . . (4)
If all initial concentrations in a batch reactor are known, it is possible, if, for example, the concentration of A is selected as the variable, to express the concentrations of the other components by means of it and the initial concentrations and thereafter to solve how the reaction proceeds in time. Likewise, in regard to a continuous flow reactor, the concentration variables on the right side of the reaction formula (3) must also be taken into consideration, but otherwise the same applies to it as to the reactor discussed above, in which the reaction was irreversible.
Reaction equilibrium. A reaction (3) tends to an equilibrium at which its gross rate is zero, i.e., it proceeds at the same rate in each direction. The equilibrium constant K of the reaction can be determined according to the equation (4) as a ratio of the kinetic velocity coefficients [3]. In the expression of the equilibrium constant, and even elsewhere in this text, concentrations are used instead of the activities of the components. The activities can be replaced with concentrations especially in dilute solutions, and concentrations can even otherwise be reconverted to activities, provided that the dependences between the two are known. In a case of gaseous components, partial pressures can be used instead of concentrations. FNT [3] Ibid., pp 12-3. ##EQU1##
The values of the equilibrium constant, which is a function of temperature, pressure and other quantities, are known for a great number of reactions and can be obtained as such from literature or through the free energy of the reaction. If a complex mixture is in equilibrium, several equilibrium equations are satisfied simultaneously. Through them and the material balances which are here considered to include also the condition of electroneutrality, the state of equilibrium is completely determined. Analogous equilibrium dependences are valid not only for chemical reactions but also for many physicochemical and physical basic processes.
In fact, all chemical reactions and most comparable physical phenomena are reversible. In some cases the equilibrium constant, and thereby the concentration ratio corresponding to the equilibrium, has a very low or very high value. In such a case an assumption can be made concerning the irreversibility of the reaction, which approximate concept has already been applied above.
Often the reaction (3) proceeds in both directions at such a high rate that the equilibrium according to the equation (5) can be considered always valid at a point of observation, and kinetics need not be taken into consideration. Such a reaction is called in this discussion a fast reaction. Some suitable examples are the reactions between many inorganic acids and bases and their soluble salts in liquid phase and the reactions between many gaseous substances at high temperatures.
In practice the appearing chemical equilibrium cannot always be expressed by means of a clear chemical equation (e.g., (3)) or a group of such equations, or such a representation is not expedient. In such a case the use of equilibrium dependences, (e.g., (5)) expressed by means of simple formulas and parameters is out of question. On the other hand they can be expressed graphically, by means of curves, tables, or otherwise, without using mathematical analytical representation. Curves and respective means can be based on information obtained from literature, or they can be determined by laboratory tests or by measurements performed in the process. Some applicable examples are many processes used in the treatment of effluents; their chemistry may not be known in detail, but an interesting concentration variable or group of variables, as well as the factors essentially affecting it, may be measured.
Well known is, for example, the unambiguous dependence of H.sup.+ ion concentration, determined as a titer curve in equilibrium conditions in a laboratory and expressed by pH, on the quantity of the added chemical, acid or base, or on the ratio of the amounts of the chemicals. This is expressed conveniently as a curve, although the necessary calculations related to the reaction of a strong acid and a strong base, for example, can also be performed analytically. Corresponding curves can be drawn up for the concentrations of other ions as well. In a case of polyvalent acids or bases or in the presence of, for example, buffering components, the curves are more complicated. If in the reaction mixture there are several components which consume the reagent, curve diagrams or corresponding representations are used in order to take the effects of the various components into consideration.
Residence-time distribution. When describing the operation of an apparatus as a mixer, the discussion is usually limited to the dependence between the inlet and the outlet channels. The residence-time distribution expresses the probability with which the particles contained in the material element entering the apparatus at a given moment can be found in the outlet channel at each later moment. A change in the inlet concentration usually does not have any effect on the operation of the process as a mixer, but the process is usually linear in regard to the concentration. The residence-time distribution can be interpreted as the weighting function of the mixing process, and thus the outlet concentration can be calculated with the aid of it, the inlet concentration varying according to an arbitrary function of time [4]. When the process parameters are constant, a convolution integral (6 ) is used for the calculation. When the flow changes, the outlet concentration can be calculated with the aid of the time-variable residence distribution (7) [4]. In literature there are examples of theoretical [2, 4] numerically known residence-time distributions [5]. FNT [2] Levenspiel, O.: Chemical reaction engineering, Wiley 1962, pp 254-9, 266, 282, 289-93. FNT [4] Niemi, A.: Tracer testing of particulate matter systems for their dynamics, in Nuclear techniques in the basic metal industries, IAEA, Vienna 1973, pp 136-9. FNT [5] Mannisto, H., Niemi, A.: On the dynamics of a cellulose bleaching plant, in Radioisotope tracers in industry and geophysics, IAEA, Vienna 1967, pp 371-84. ##EQU2## g residence-time distribution (weighting function) C.sub.i concentration of component i in the outlet channel
The residence-time distribution/weighting function can be used to describe linear processes only. When kinetic reactors are considered, the residence-time distribution/weighting function can thus generally be used as a basis of the consideration of only a reactor in which a reaction of first-order kinetics occurs [2]. FNT [2] Levenspiel, O.: Chemical reaction engineering, Wiley 1962, pp 254-9, 266, 282, 289-93.
In special cases of mixing it has been possible to consider even other than first order reactions simultaneously with the mixing phenomenon. The reactors that can be considered in such cases are mainly only the plug-flow reactor and the stirred tank reactor with perfect mixing. In a plug-flow reactor the material elements entering simultaneously are considered to become immediately mixed with each other, but mixing does not occur between elements entering at different times. Since all elements have the same residence time or residence parameter value, the reactor does not have an actual residence-time distribution but this has degenerated into a constant delay time. Such a reactor can be considered substantially in the same manner as a batch reactor. An ideal mixing reactor, for its part, can be described by using an ordinary non-linear first-order differential equation or equation group, which can usually be solved at least numerically. The known functional form of the residence-time distribution is thus not utilized in connection with the solution.
Control: When a reactor is controlled in order to reach the desired outlet concentration, conventional feedback control is usually applied in practice. For reasons of measurement, the most common object of control is the pH, i.e., the H.sup.+ (OH.sup.-) ion concentration, but even other concentration quantities are controlled in connection with continuous flow processes. Feedforward control is also applied in practice with the aim of compensating changes in the concentrations at the time they enter the process. The process models used in such cases are usually very simple. The follow-up control of combustion air in a constant proportion to the fuel feed is a suitable example of the control of the combustion reaction product, even though this follow-up control is not based on measuring the concentration. Sometimes a relatively simple feedforward circuit is used for pH control in connection with feedback control (see, e.g., [6]). The chemicals feed to a flotation process are controlled in some concentration plants so that they are in a constant proportion to the concentration of the incoming ore or the feed of the mineral component to be concentrated. FNT [6] Shinskey, F. G., Myron, T. J.: Adaptive feedback applied to feedforward pH control, in Advances in instrumentation 25, Paper No. 565-70, ISA, Pittsburgh 1970.
The analytical consideration of reactor dynamics and control has especially related to reactors with first-order kinetics. The ideal mixing reactor of the first order has been the principal object of interest (see, e.g., [7]). An exothermal ideal mixing reactor with first-order kinetics which is non-linear owing to the temperature dependence of the reaction rate coefficient has also been the object of keen theoretical interest [8]. Feedforward control of a first-order flotation reactor, using a model coupled parallelly to the process, has been suggested [9]. An example of optimal control of such a reactor is contained in, for example, [10]. FNT [7] Solheim, O. A.: A guide to controlling cascaded chemical reactors, Control Engineering, July 1961, pp 79-85. FNT [8] Oppelt, N., Wicke, E. (Ed.): Grundlagen der chemischen Prozessregelung, Oldenbourg 1964, pp 46-125. FNT [9] Niemi, A.: A study of dynamic and control properties of industrial flotation processes, Acta Polyt. Scand. Chem. No. 48, Helsinki 1966, pp 47-8. FNT [10] Niemi, A., Maijanen, J., Nihtila, M.: Singular optimal feedforward control of flotation, IFAC Symp, on Optimization Methods, Varna, Bulgaria 1974-10-08 . . . 11.
Control of fast reactors in which the kinetics of the reactions can be left without consideration has in some cases been considered analytically. Thus, in [11] a method has been suggested for the control of a non-linear plug-flow reactor. Feedback control of an ideal stirred tank reactor in which likewise a fast non-linear reaction occurs is analyzed in [12, 13], and compared with an optimal control from a given initial state to the origin which problem is rarely encountered in practice, in [14]. In particular, poor mixing and the presence of buffering chemicals generally complicate obtaining acceptable results in, for example, pH control [6, 12 with references]. A satisfactory, generally applicable method for the consideration of a non-linear reactor with arbitrary flow characteristics and the control of the reactor has not yet been introduced. FNT [6] Shinskey, F. G., Myron, T. J.: Adaptive feedback applied to feedforward pH control, in Advances in instrumentation 25, Paper No. 565-70, ISA, Pittsburgh 1970. FNT [11] Talonen, T., Niemi, A.: Modelling of a pyrite smelting process, 4. IFAC Congress, Warsaw 1969, Session No. 66, pp 141-55. FNT [12] Orava, J., Niemi, A.: State model and stability analysis of a pH control process, Int. J. of Control 20 (1974), pp 557-67. FNT [13] Richter, J. D., Fournier, C. D., Ash, R. H., Marcikic, S.: Waste neutralization control, Instr. Technology 21 (1974) 4, pp 35-40. FNT [14] McAvoy, Th.J.: Time optimal and Ziegler-Nichols control, Ind. & Eng. Chemistry, Process Des. & Dev. 11 (1972), pp 71-78.